Volterra Equations Of Convolution Type Research Articles

A weak solution theory for stochastic Volterra equations of convolution type

We obtain general weak existence and stability results for stochastic convolution equations with jumps under mild regularity assumptions, allowing for non-Lipschitz coefficients and singular kernels. Our approach relies on weak convergence in Lp spaces. The main tools are new a priori estimates on Sobolev–Slobodeckij norms of the solution, as well as a novel martingale problem that is equivalent to the original equation. This leads to generic approximation and stability theorems in the spirit of classical martingale problem theory. We also prove uniqueness and path regularity of solutions under additional hypotheses. To illustrate the applicability of our results, we consider scaling limits of nonlinear Hawkes processes and approximations of stochastic Volterra processes by Markovian semimartingales.

Linear-quadratic control for a class of stochastic Volterra equations: Solvability and approximation

We provide an exhaustive treatment of linear-quadratic control problems for a class of stochastic Volterra equations of convolution type, whose kernels are Laplace transforms of certain signed matrix measures which are not necessarily finite. These equations are in general neither Markovian nor semimartingales, and include the fractional Brownian motion with Hurst index smaller than 1/2 as a special case. We establish the correspondence of the initial problem with a possibly infinite dimensional Markovian one in a Banach space, which allows us to identify the Markovian controlled state variables. Using a refined martingale verification argument combined with a squares completion technique, we prove that the value function is of linear quadratic form in these state variables with a linear optimal feedback control, depending on nonstandard Banach space valued Riccati equations. Furthermore, we show that the value function of the stochastic Volterra optimization problem can be approximated by that of conventional finite dimensional Markovian linear-quadratic problems, which is of crucial importance for numerical implementation.

Paracontrolled distribution approach to stochastic Volterra equations

Based on the notion of paracontrolled distributions, we provide existence and uniqueness results for rough Volterra equations of convolution type with potentially singular kernels and driven by the newly introduced class of convolutional rough paths. The existence of such rough paths above a wide class of stochastic processes including the fractional Brownian motion is shown. As applications we consider various types of rough and stochastic (partial) differential equations such as rough differential equations with delay, stochastic Volterra equations driven by Gaussian processes and moving average equations driven by Lévy processes.

Solvability and uniform local attractivity for a Volterra equation of convolution type

We show the existence of uniformly locally attractive solutions for a nonlinear Volterra integral equation of convolution type with a general kernel. We use methods and techniques of fixed point theorems and properties of measure of noncompactness. We extend earlier results obtained in the context of integral equations of fractional order. We give new insights about a new and striking relation between the size of data and the fractional order $\alpha >0$ of the kernel $k(t)=t^{\alpha -1}/\Gamma (\alpha )$.

Application of measure of noncompactness to Volterra equations of convolution type

Sufficient conditions for the existence of at least one solution of a nonlinear integral equation with a general kernel are established. The existence result is proved in $C([0,T],E)$, where $E$ denotes an arbitrary Banach space. We use the Darbo-Sadovskii fixed point theorem and techniques of measure of noncompactness. We extend and generalize results obtained by other authors in the context of fractional differential equations. One example illustrates the theoretical results.

Fractional stochastic Volterra equation perturbed by fractional Brownian motion

In this paper, we consider a class of fractional stochastic Volterra equation of convolution type driven by infinite dimensional fractional Brownian motion with Hurst index h∈(0,1). Base on the explicit formula for the scalar resolvent function and the properties of the Mittag–Leffler’s function, the existence and regularity results of the stochastic convolution process are established. Separate proofs are required for the cases of Hurst parameter above and below 12 and it will turn out that the regularity of the solution increases with Hurst parameter h.

A Series Approach to Perturbed Stochastic Volterra Equations of Convolution Type

In the paper, perturbed stochastic Volterra Equations with noise terms driven by series of independent scalar Wiener processes are considered. In the study, the resolvent approach to the equations under consideration is used. Sufficient conditions for the existence of strong solution to the class of perturbed stochastic Volterra Equations of convolution type are given. Regularity of stochastic convolution is supplied, as well.

Efficient resolution of metastatic tumor growth models by reformulation into integral equations

The McKendrick/Von Foerster equation is a transport equation with a non-local boundary condition that appearsfrequently in structured population models. A variant of this equation with a size structure has beenproposed as a metastatic growth model by Iwata et al.  &nbspHere we will show how a family of metastatic models with 1D or 2D structuring variables, based on the Iwata model, can bereformulated into an integral equation counterpart, a Volterra equation of convolution type, for which a rich numerical andanalytical theory exists. Furthermore, we will point out the potential of this reformulation by addressing questions coming up inthe modelling of metastatic tumour growth. We will show how this approach permits to reduce the computational costof the numerical resolution and to prove structural identifiability.

Stochastic Volterra equations under perturbations

We study stochastic perturbed Volterra equations of convolution type in an infinite dimensional case. Our interest is directed towards the existence and regularity of stochastic convolutions connected to the equations considered under some kind of perturbations. We use an operator theoretical method for the representation of solutions.

Stability and asymptotic properties of a linear fractional difference equation

This paper discusses qualitative properties of the two-term linear fractional difference equation where , , and is the α th order Riemann-Liouville difference operator. For this purpose, we show that this fractional equation is the Volterra equation of convolution type. This enables us to analyse its qualitative properties by use of tools standardly employed in the qualitative investigation of Volterra difference equations. As the main result, we derive a sharp condition for the asymptotic stability of the studied equation and, moreover, give a precise asymptotic description of its solutions. MSC:39A30, 39A12, 26A33.

Numerical Solutions to Fractional Perturbed Volterra Equations

In the paper, a class of perturbed Volterra equations of convolution type with three kernel functions is considered. The kernel functions , , , correspond to the class of equations interpolating heat and wave equations. The results obtained generalize our previous results from 2010.

Regularity of solutions to stochastic Volterra equations of convolution type

In this paper we consider two classes of linear Volterra equations: of convolution type on ℝ+ and integro-differential with infinite delay on d-dimensional torus, both driven by spatially homogeneous Wiener process. First, we study the existence of solutions to these equations in the space of tempered distributions and then derive conditions under which the solutions take values in Sobolev spaces. We give necessary and sufficient conditions providing regularity of solutions to equations considered. The harmonic analysis techniques and stochastic integration in function spaces are used. The paper is a short review of regularity results for stochastic Volterra equations of convolution type.

Singular perturbation for Volterra equations of convolution type

Under the assumption that A is the generator of a twice integrated cosine family and K is a scalar valued kernel, we solve the singular perturbation problem ( E ϵ ) ϵ 2 u ϵ ″ ( t ) + u ϵ ′ ( t ) = Au ϵ ( t ) + ( K ∗ Au ϵ ) ( t ) + f ϵ ( t ) , ( t ⩾ 0 ) ( ϵ > 0 ) , when ϵ → 0 +, for the integrodifferential equation ( E) w ′ ( t ) = Aw ( t ) + ( K ∗ Aw ) ( t ) + f ( t ) , ( t ⩾ 0 ) , on a Banach space. If the kernel K verifies some regularity conditions, then we show that problem ( E ϵ ) has a unique solution u ϵ ( t) for each small ϵ > 0. Moreover u ϵ ( t) converges as ϵ → 0 +, to the unique solution u( t) of equation ( E).

Periodic solutions of discrete Volterra equations

In this paper, we investigate periodic solutions of linear and nonlinear discrete Volterra equations of convolution or non-convolution type with unbounded memory. For linear discrete Volterra equations of convolution type, we establish Fredholm’s alternative theorem and for equations of non-convolution type, and we prove that a unique periodic solution exists for a particular bounded initial function under appropriate conditions. Further, this unique periodic solution attracts all other solutions with bounded initial function. All solutions of linear discrete Volterra equations with bounded initial functions are asymptotically periodic under certain conditions. A condition for periodic solutions in the nonlinear case is established.

On the inversion of the Laplace transform for resolvent families in UMD spaces

We analyze the inversion of the Laplace transform in UMD-spaces for resolvent families associated to an integral Volterra equation of convolution type.

Rates of approximation and ergodic limits of regularized operator families

We state mean ergodic theorems with rates of approximation for a new class of operator families. Our result provides a unified approach to convergence rates for many particular strongly continuous solution families associated to linear evolution equations such as the abstract Cauchy problem of the first and second order, and integral Volterra equations of convolution type. We discuss in particular, applications to α-times integrated cosine families, k-convoluted semigroups and integral resolvents.

Mittag-Leffler's Function and Stochastic Linear Volterra Equations of Convolution Type

We consider the class of stochastic linear Volterra equations of convolution type defined by fractional integration kernels g ρ (t)={ 1 Γ(ρ) }t ρ−1, ρ∈(0,2). Using an explicit formula for the scalar resolvent function, we establish the basic properties of the stochastic convolution process W S . Our formulas are given in terms of the Mittag-Leffler's function

Estimation of the solutions of Volterra difference equations

Volterra equations, whose solution is defined by the whole previous history, are widely used in the modelling of the processes in continuous mechanics and biomechanics, problems of control and estimation and also some schemes of numerical solutions of integral and integral–differential equations [1–8]. In this connection, there is an essential interest in such properties of the solutions as stability, limiting periodicity, boundedness and various estimates of the solutions defined by the acting perturbations. In [9], such estimates were obtained for the solutions of nonlinear scalar integral Volterra equations of convolution type

Regularized Solutions for Abstract Volterra Equations

We introduce a two-kernel dependent family of strong continuous operators defined in a Banach space, which allows us to consider in an unified treatment the notions of, among others, C0-semigroups of operators, cosine families, n-times integrated semigroups, resolvent families and k-generalized solutions.The results are applied to the study of existence and uniqueness of solutions for the Volterra equation of convolution type u=f+a∗Au, in the case A is not necessarily densely defined. Examples for equations defined in Lp spaces are also given.

Uniform continuity and compactness for resolvent families of operators

We characterize the uniform continuity and the compactness of a resolvent family of operators {R(t)t⩾0 for a Volterra equation of convolution type denned in a Banach spaceX. In particular, we extend similar results to those for semigroups of operators and cosine families of operators studied in other works.